Franz Gähler, John Hunton and Johannes Kellendonk

Abstract

We study the cohomology and hence
K–theory of
the aperiodic tilings formed by the so called “cut and project” method, that is, patterns in
d–dimensional
Euclidean space which arise as sections of higher dimensional, periodic structures.
They form one of the key families of patterns used in quasicrystal physics, where
their topological invariants carry quantum mechanical information. Our work
develops both a theoretical framework and a practical toolkit for the discussion and
calculation of their integral cohomology, and extends previous work that only
successfully addressed rational cohomological invariants. Our framework unifies the
several previous methods used to study the cohomology of these patterns. We
discuss explicit calculations for the main examples of icosahedral patterns in
ℝ3 – the
Danzer tiling, the Ammann–Kramer tiling and the Canonical and Dual Canonical
D6
tilings, including complete computations for the first of these, as well as results for
many of the better known 2–dimensional examples.